Basic Probability & Random Variables

Axioms of Probability

• If are mutually disjoint, then

Conditional Probability

𝑆

𝐴

𝐵

𝐴∩𝐵

Multiplicative Rule of Probability

BAYES RULE:

Bayes Rule is very important

Why?• Often we want to know • But what we do know is • We will be able to infer by

Here is a useful application of Bayes

From the graph we can see that,

𝑆

𝐴

𝐵

𝐴∩ 𝐵

Randomization Response Theory

Assume that you need estimate the proportion of narcotic drug consumption among university students. It is unlikely that students would answer your questionnaire honestly. So here is a simple trick you may use.

Instead of asking the question directly, let the student draw a ball from an urn in which there are 8 blue and 2 yellow balls. If a yellow ball is drawn (you do not see the result), student answers the question «Is the last digit of your TC ID number odd?» and if a blue ball is drawn then the student answers «Have you ever used a narcotic drug?» question.

• Assume you have asked this question to 17 students and 13 of them answered YES.

Soon we will be able to compute the error due to …(?)

Random Variables and Probability Distributions

• Random Variable:?

Example: When rolling a two dice, we may be interested in whether or not the sum of the two dice is 7. Or we might be interested in the sum of the two dice.

Example:How long does it take for the next bus to arrive?

• Now, suppose the probability that the T comes in any given minute is a constant , and whether the T comes is independent of what has happened in previous periods.

• What's P(X=1)?

• What's P(X=2)?

• What’s P(X=3)?

• What’s P(X=x)?

Geometric Distribution with a parameter

Probability Density Function

• An alternative model where Y is exact time:

If In class: How probabilities are related with areas under the curve.

Expectation

• Discrete Case

• Continuous Case

• Discrete

• Continuous

Variance

• Typically you need to know what sort of probability distributions are there and for which type of situations thay are used for.

• We will be mostly dealing with Normal Distribution.

INCOME DISTRIBUTION – (Empirical)

INCOME DISTRIBUTION – (Theoretical)

Log

Nor

mal

Did

trib

ution

Normal Distribution• Normal distribution has an unfriendly form

that does not let explicit integration:

• However any normal distribution can be transformed into standard normal distribution

Normal Distribution

xm

s

m=0

s=1

z

Standard Normal Distribution-xz m

s=

P(x < 500) = P(z < 1)

Normal Distribution

600μ =500

P(x < 600)

μ = 500 σ = 100

x

Standard Normal Distribution

600 500 1100

xz ms

= = =

1μ = 0

μ = 0 σ = 1

z

P(z < 1)

Same Area

• Before going any further did you notice that statistical parameters are actually operational definitions for some concepts.

• Let’s discuss these operationalized variables and their corresponding concepts:

Sampling

Probability Sampling Nonprobability Sampling

Probability Sampling

• Sampling element• Population• Target population• Sampling frame• Sampling ratio

There is a classic Jimmy Stewart movie, Magic Town, about "Grandview," a small town in the Midwest that is a perfect statistical microcosm of the United States, a place where the citizens' opinions match perfectly with Gallup polls of the entire nation. A pollster (Jimmy Stewart), secretly uses surveys from this "mathematical miracle" as a shortcut to predicting public opinion. Instead of collecting a national sample, he can more quickly and cheaply collect surveys from this single small town. The character played by Jane Wyman, a newspaper editor, finds out what is going on and publishes her discovery. As a result the national media descend upon the town, which becomes, overnight, "the public opinion capital of the U.S."

Probability Sampling

• Check http://www.socialresearchmethods.net

POPULATION PARAMETERS SAMPLE STATISTICS

To b

e fil

led

in c

lass

Sampling Distribution

Probability Sampling

• Random sample• Sampling error

• Four Ways to Sample Randomly– Simple Random– Systematic– Stratified Sampling– Cluster Sampling

Random Sample

• Sampling Error:

𝑥=0.5

𝜇=0.5625

𝑥=0.75

Variation Component

Sample size Component

Sampling Distribution and Sampling Error

Let’s first see what mathematics have to say.

1. According to Law of Large Numbers:

As sample size increases (approaches to ) sample mean approaches to population mean, in mathematical symbols

2. According to Central Limit Theorem

As the number of samples (not the sample size, this time) increases then sample mean has a normal distribution with mean and standard deviation . Mathematically we say,

Sampling and Confidence

x

𝜇 𝑥𝑢𝑥 𝑙 𝑥

𝑥−(𝑧∗ 𝜎√𝑛 )≤𝜇 ≤𝑥+(𝑧∗ 𝜎

√𝑛 )1. Confidence information is in z.2. can be replaced by .